Prove cos^2t+4cost+4/cost+2=2sect+1/sect
To prove this equation, we need to simplify the left-hand side (LHS) and the right-hand side (RHS) of the equation separately and show that they are equal.
Let's start with the LHS:
cos^2t + 4cos(t) + 4 / cos(t) + 2
Step 1: Factor the numerator
(cos^2t + 2cos(t) + 2cos(t) + 4) / (cos(t) + 2)
Note: Here, we have split the term 4cos(t) into 2cos(t) + 2cos(t) for easier factoring.
Step 2: Rearrange the terms in the numerator
[(cos^2t + 2cos(t)) + (2cos(t) + 4)] / (cos(t) + 2)
Step 3: Factor the first two terms in the numerator
[cos(t)(cos(t) + 2) + 2cos(t) + 4] / (cos(t) + 2)
Step 4: Combine similar terms in the numerator
[(cos(t) + 2)(cos(t) + 2)] / (cos(t) + 2)
Step 5: Simplify the numerator
= (cos(t) + 2)
Now let's simplify the RHS:
2sec(t) + 1 / sec(t)
Step 1: Rewrite sec(t) in terms of cos(t)
= 2(1 / cos(t)) + 1 / (1 / cos(t))
Step 2: Simplify the expression
= 2 / cos(t) + cos(t)
Step 3: Find a common denominator and combine the terms
= (2 + cos^2(t)) / cos(t)
Step 4: Rewrite cos^2(t) in terms of (cos(t) + 2)
= [(cos(t) + 2) + (cos^2(t) - 2)] / cos(t)
Step 5: Simplify the numerator
= (cos(t) + 2)
Now, we can see that both the LHS and RHS equal (cos(t) + 2). Hence, we can conclude that cos^2t + 4cost + 4/cost + 2 = 2sect + 1/sect is true.